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Math Problems
Algebra 2
Simplify the product of two radical expressions having same variable
Select all the rational numbers.
\newline
Multi-select Choices:
\newline
(A)
6
6
6
\newline
(B)
−
13
-13
−
13
\newline
(C)
144
\sqrt{144}
144
\newline
(D)
8
3
\sqrt[3]{8}
3
8
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Select all the rational numbers.
\newline
Multi-select Choices:
\newline
(A)
−
2
-2
−
2
\newline
(B)
2
\sqrt{2}
2
\newline
(C)
9
\sqrt{9}
9
\newline
(D)
64
3
\sqrt[3]{64}
3
64
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Examples
\newline
(a)
−
32
5
\sqrt[5]{-32}
5
−
32
\newline
(b)
−
32
5
-\sqrt[5]{32}
−
5
32
\newline
(c)
81
4
\sqrt[4]{81}
4
81
\newline
(d)
81
\sqrt{81}
81
\newline
(e)
−
32
\sqrt{-32}
−
32
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4
x
+
1
3
=
2
8
x
\sqrt[3]{4^{x+1}}=2 \sqrt{8^{x}}
3
4
x
+
1
=
2
8
x
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c)
2
7
2
x
−
1
3
x
+
1
3
=
9
\sqrt[3]{\frac{27^{2 x-1}}{3^{x+1}}}=9
3
3
x
+
1
2
7
2
x
−
1
=
9
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Solve for
d
d
d
. Express your answer in simplest radical form if necessary.
\newline
d
=
123.3
3
⋅
123.3
3
⋅
123.3
3
d=\sqrt[3]{123.3} \cdot \sqrt[3]{123.3} \cdot \sqrt[3]{123.3}
d
=
3
123.3
⋅
3
123.3
⋅
3
123.3
\newline
Answer:
d
=
d=
d
=
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Solve for
b
b
b
. Express your answer in simplest radical form if necessary.
\newline
b
3
=
31
b^{3}=31
b
3
=
31
\newline
Answer:
b
=
b=
b
=
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Solve for
x
x
x
. Express your answer in simplest radical form if necessary.
\newline
x
2
=
9
x^{2}=9
x
2
=
9
\newline
Answer:
x
=
x=
x
=
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Solve for
a
a
a
. Express your answer in simplest radical form if necessary.
\newline
a
=
115.3
3
⋅
115.3
3
⋅
115.3
3
a=\sqrt[3]{115.3} \cdot \sqrt[3]{115.3} \cdot \sqrt[3]{115.3}
a
=
3
115.3
⋅
3
115.3
⋅
3
115.3
\newline
Answer:
a
=
a=
a
=
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Find equivalent to
81
a
4
b
8
4
\sqrt[4]{81 a^{4} b^{8}}
4
81
a
4
b
8
?
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r
5
r
7
6
\sqrt{r^{5}} \sqrt[6]{r^{7}}
r
5
6
r
7
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1
1
1
.
1
4
n
3
−
3
8
n
5
\frac{1}{4} n^{3}-\frac{3}{8} n^{5}
4
1
n
3
−
8
3
n
5
\newline
Standard form:
\newline
Degree:
\newline
Leading coefficient:
\newline
Classify the polynomial:
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Which exponential expression is equivalent to
t
6
\sqrt[6]{t}
6
t
?
\newline
Choose
1
1
1
answer:
\newline
(A)
t
6
t^{6}
t
6
\newline
(B)
1
t
1
6
\frac{1}{t^{\frac{1}{6}}}
t
6
1
1
\newline
(C)
t
1
6
t^{\frac{1}{6}}
t
6
1
\newline
(D)
1
t
6
\frac{1}{t^{6}}
t
6
1
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1
0
6
(
1
0
6
+
1
)
(
1
0
6
+
2
)
(
1
0
6
+
3
)
+
1
\sqrt{10^{6}\left(10^{6}+1\right)\left(10^{6}+2\right)\left(10^{6}+3\right)+1}
1
0
6
(
1
0
6
+
1
)
(
1
0
6
+
2
)
(
1
0
6
+
3
)
+
1
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−
243
m
3
3
\sqrt[3]{-243 m^{3}}
3
−
243
m
3
=
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Which expressions are equivalent to
(
d
4
)
3
(\sqrt[4]{d})^{3}
(
4
d
)
3
?
\newline
Choose all answers that apply:
\newline
A
d
3
4
\sqrt[4]{d^{3}}
4
d
3
\newline
B
(
d
1
4
)
3
\left(d^{\frac{1}{4}}\right)^{3}
(
d
4
1
)
3
\newline
c
d
4
3
\sqrt[3]{d^{4}}
3
d
4
\newline
D None of the above
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Which expressions are equivalent to
(
d
4
)
3
(\sqrt[4]{d})^{3}
(
4
d
)
3
?
\newline
Choose all answers that apply:
\newline
(A)
d
3
4
\sqrt[4]{d^{3}}
4
d
3
\newline
(B)
d
4
3
\sqrt[3]{d^{4}}
3
d
4
\newline
(C)
(
d
1
4
)
3
\left(d^{\frac{1}{4}}\right)^{3}
(
d
4
1
)
3
\newline
(D) None of the above
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Write the expression in exponential form.
\newline
x
3
5
\sqrt[5]{x^3}
5
x
3
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Simplify the expression completely.
\newline
64
3
+
−
8
3
−
5
125
3
−
216
3
\sqrt[3]{64}+\sqrt[3]{-8}-5 \sqrt[3]{125}-\sqrt[3]{216}
3
64
+
3
−
8
−
5
3
125
−
3
216
\newline
Answer:
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For the function
f
(
x
)
=
(
4
x
)
3
f(x)=\sqrt[3]{(4 x)}
f
(
x
)
=
3
(
4
x
)
, find
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
.
\newline
f
−
1
(
x
)
=
(
x
4
)
3
f^{-1}(x)=\left(\frac{x}{4}\right)^{3}
f
−
1
(
x
)
=
(
4
x
)
3
\newline
f
−
1
(
x
)
=
(
4
x
)
3
f^{-1}(x)=(4 x)^{3}
f
−
1
(
x
)
=
(
4
x
)
3
\newline
f
−
1
(
x
)
=
x
3
4
f^{-1}(x)=\frac{x^{3}}{4}
f
−
1
(
x
)
=
4
x
3
\newline
f
−
1
(
x
)
=
4
x
3
f^{-1}(x)=4 x^{3}
f
−
1
(
x
)
=
4
x
3
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Given
x
>
0
x>0
x
>
0
, the expression
x
21
10
\sqrt[10]{x^{21}}
10
x
21
is equivalent to
\newline
x
x
10
x \sqrt[10]{x}
x
10
x
\newline
x
2
x
2
10
x^{2} \sqrt[10]{x^{2}}
x
2
10
x
2
\newline
x
2
x
10
x^{2} \sqrt[10]{x}
x
2
10
x
\newline
x
x
2
10
x \sqrt[10]{x^{2}}
x
10
x
2
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Assuming
x
x
x
and
y
y
y
are both positive, write the following expression in simplest radical form.
\newline
y
3
28
x
4
y
6
y^{3} \sqrt{28 x^{4} y^{6}}
y
3
28
x
4
y
6
\newline
Answer:
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Assuming
x
x
x
and
y
y
y
are both positive, write the following expression in simplest radical form.
\newline
3
28
x
2
y
3
3 \sqrt{28 x^{2} y^{3}}
3
28
x
2
y
3
\newline
Answer:
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Select the equivalent expression.
\newline
(
u
−
1
u
8
)
5
6
\left(\frac{u^{-1}}{u^{8}}\right)^{\frac{5}{6}}
(
u
8
u
−
1
)
6
5
\newline
u
15
\sqrt{u^{15}}
u
15
\newline
1
u
15
\frac{1}{\sqrt{u^{15}}}
u
15
1
\newline
u
2
15
\sqrt[15]{u^{2}}
15
u
2
\newline
1
u
2
15
\frac{1}{\sqrt[15]{u^{2}}}
15
u
2
1
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7
w
−
5
4
2
=
w
−
2
4
\frac{\sqrt[4]{7w-5}}{2}=\sqrt[4]{w-2}
2
4
7
w
−
5
=
4
w
−
2
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Let
y
=
e
x
y=\sqrt{e^{x}}
y
=
e
x
.
\newline
Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Choose
1
1
1
answer:
\newline
(A)
e
x
2
\frac{\sqrt{e^{x}}}{2}
2
e
x
\newline
(B)
e
x
\sqrt{e^{x}}
e
x
\newline
(C)
x
e
x
−
1
x \sqrt{e^{x-1}}
x
e
x
−
1
\newline
(D)
1
2
e
x
\frac{1}{2 \sqrt{e^{x}}}
2
e
x
1
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